A review of generalized inverse matrix and its block structure

Subject Areas : Statistics

fatemeh babakordi ¹

1 - Assistant Professor, Department of Mathematics and statistics, Faculty of Basic Science, Gonbad Kavous University, Gonbad Kavous, Iran.

Received: 2020-01-05 Accepted : 2020-04-09 Published : 2021-05-22

Keywords: ماتریس منفرد, ماتریس نامنفرد, نمایش بلوکی, معکوس تعمیم یافته,

Abstract :

If a matrix is square and nonsingular namely its rows (or columns) are linear independent, it is said that the matrix is invertible. In recent years, in various fields of applied mathematics, there has been a need to find the inverse of singular and rectangular matrices. Therefore, an inverse was defined that founds the inverse for larger class of nonsingular matrices while still having some properties of conventional inverse which gives the same inverse when the matrix is nonsingular. This inverse is called generalized inverse or quasi-inverse. In this paper, the most common types of such inverse are investigated.

References:

[1] Y. Akatsuka and T. Matsuo, Optimal control of linear discrete systems using the generalized inverse of a matrix, Techn Rept. 13, Institute of Automatic Control, Nagoya Univ., Nagoya, Japan (1965)

 

[2] V. Aleksi´c and V. Rakoˇcevi´c, Approximate properties of the Moore-Penrose inverse, VIII Conference on Applied Mathematics (Tivat, 1993), Univ. Montenegro, Podgorica 1–14 (1994)

 

[3] E. Arghiriade and A. Dragomir, Une nouvelle definition de l’inverseg´en´eralise´ed’unematrice, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat.Natur 8: 158–165 (1963)

 

[4] R. B. Bapat, Generalized inverses with proportional minors, Linear Algebra and its Applications 211:27-33(1994)

 

[5] C. Badea and M. Mbekhta, Generalized inversesand the maximal radius of regularity of a Fredholm operator, Integral Equations Operator Theory 28: 133–146(1997)

[6] N. Castro Gonz´alez, J. J. Koliha, and YiminWeiPerturbation of the Drazin inverse for matrices with equal eigenprojections at zero, Linear Algebra and its Applications 312:181-19 (2000)

 

 

[7] V. Miler Jerkovi´c, B. Maleševi´c, Block representation of generalized inverses of matrices, in: Proceedings of the fifth Symposium “Mathematics and applications”, Faculty of Mathematics, University of Belgrade and Serbian Academy of Sciences and Arts 1:176-185(2014)

 

[8] R. Penrose. A generalized inverses for matrices. Mathematical Proceedings of the Cambridge Philosophical Society 15:406-413(1955)

 

[9] C.A. Rhode. Contribution to the theory, computation and application of generalized inverses (PhD dissertation). University of North Carolina at Releigh (1964)

 

[10] S.L. Cambell, C.D. Meyer. Inverses of Linear Transformations. Siam, Philadelphia (2009)

 

[11] M. Nikuie, Singular fuzzy linear systems, Appl. Math. Comput. Intell 2: 157-168(2013)

 

[12] B. Radičić, B. Malešević. Some considerations in Relation to the Matrix Equation AXB=C. The Mediterranean Journal of Mathematics 11:841-856  (2014)

 

[13] V. Miler Jerkovi´c, B. Mihailovi´c, B. Maleševi´c, A new method for solving square fuzzy linear systems, in: J. Kacprzyk, E. Szmidt, S. Zadrozny,  K. Atanassov, M. Krawczak (Eds.), Advances in Fuzzy Logic and Technology 2017. IWIFSGN 2017, EUSFLAT 2017, in: Adv. Intell. Syst. Comput., vol. 642, Springer, Cham 278-280(2017)

 

[14] V. Miler Jerkovi´c, B. Maleševi´c, Block representation of generalized inverses of matrices, in: Proceedings of the fifth Symposium “Mathematics and applications”, vol. 1, Faculty of Mathematics, University of Belgrade and Serbian Academy of Sciences and Arts 176-185(2014)

 

[15] A. Ben-Israel, T.N.E. Greville, Generalized Inverses, Theory and Applications, Springer, New York  (2003)

 

[16] B. J. Maleševi´c, Grupnafunkcionalnajednaˇcina, Magister thesis, University of Belgrade (1998)

 

[17] B. Mihailovic, Vera Miler Jerkovic, BrankoMaleševic, Solving fuzzy linear systems using a block representation of generalized inverses: The group inverse, Fuzzy Sets and Systems 353:66-85(2018)